Metric Modulation

Metric modulation is the changing of a time signature or tempo. Both are used in progressive music and add variety and originality to the music. When changing time signatures, the tempo should stay the same. A change in time signatures must sound as smooth as possible.
1a. Simple Metric Modulations
Simple metric modulation is when the time signature changes stay consistent and structured. Here are some simple and common examples to get started with. The first example starts in 4/4 and modulates to 3/4 for one measure, then goes back to 4/4. This kind of modulation can be found in almost every kind of music, even rap. The last two measures could be combined to make 7/4 (because 3+4=7) but using four measures is just simpler.
midi sample (Simple Metric Modulation)

*The L in the last measure means the note is sustained for the duration indicated... in this case a dotted quarter note.
The next example alternates between 3/4 and 5/8 each bar. Modulating like this may be hard and confusing because of the beat change.

1b. Complex Metric Modulation
When modulating between 3 or more different time signatures, it is referred to as complex metric modulation. Complex metric modulation can be structured (by following a repeating pattern or idea) or unstructured and spontaneous. A rather overlooked sub genre of progressive rock is "Math Rock" (or Tech Metal/Math metal/Mathcore etc.) Math rock, as it's name suggests, is a form or progressive rock that uses math as a musical outlet for creativity. Math rock puts a very untraditional, sometimes akward spin on standard rock/punk/acustic music by using complex metric modulation and rhythm patterns. Some of the more extreme math rock bands won't even think about using 4/4 or 3/4, but will use 13/8, 17/4, 11/8 and other crazy time signatures. Sometimes math rock will use odd grouping with meters, like 3, 5, or 7 bar riffs, rather than the traditional 4, 2, or 8 bars. This style of progressive music is very hard to write in because it sounds very unfamiliar to our ears. Some of the more popular math rock bands are Don Caballero, Breadwinner, and Bozart.
The example below is the introduction to a song I wrote. It uses elements of math rock and progressive rock by changing time signatures after every bar. The intorduction is 28 bars long, so I figured I would give a picture instead of actual text tablature. It will help to figure out the counting on your own.
midi sample (The Sky Is Missing)
jpeg files (The Sky Is Missing. Page 1 | Page 2)
1c. Mathematical Application To Time Signatures
Time signatures can be figured out in many ways. My favorite way is to think of them in terms of quarter notes. For example, 4/4 has 4 quarter notes in a measure. 3/4 has 3 quarter notes in a measure. 5/4 has 5 and 6/4 has 6. Now, if you look at a time signature like 7/8, you know that there are 7 eighth notes in a beat. Since 2 eighth notes = 1 quarter note, that would mean there are 3.5 quarter notes per measure. Look at it this way:

You can figure out how many quarter notes are in any x/8 time signature using this method.
1d. Practicing/Composing
A good way to practice metric modulation is by using a metronome. Set the metronome to about 60bpm and play through a few of your original riffs in 4/4. Now take these riffs and add a few notes to them (5/4), then subtract (3/4). Finally, try to make a smooth transition between the riffs. Try doing this with other time signatures. Remember, it is very important that the tempo does not change when using/ practicing metric modulation. Here is an example that goes from 9/8 to 11/8 to 9/8 to 4/4 using the add/subtract idea:
midi sample (Metric Modulation Practice)

Another way of practicing is to count beats in your head. This can be done without your guitar so you can do it in a car, in school, wherever. You do this by modulating in your head with numbers. For example, you can count eighth notes and modulate between 4/4 and 7/8:

1, 2, 3, 4, 5, 6, 7, 8 - 1, 2, 3, 4, 5, 6, 7

or, 3/4 and 5/8

1, 2, 3, 4, 5, 6 - 1, 2, 3, 4, 5

or, 5/4 and 4/4

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 - 1, 2, 3, 4, 5, 6, 7, 8

The possibilities are endless. Make sure to keep a steady tempo when doing this exercise. After you get good with eighth notes, try sixteenth notes, quarter notes, triplets, etc. I don't recommend mixing note values during mental practice because that may get too confusing.

Chord Progressions And Harmonic Concepts

Chord progressions in progressive music can go in almost any direction. The more chords used, the more complex the song will be. You could write a song where no chord is ever repeated, or a song with just a few diatonics. In progressive music, non-diatonic chords are used to make harmony more interesting.
2a. Diatonic And Non-Diatonic Chords
Basically, a diatonic progression is when all chords belong to the key. Here is the C major scale:

I ii iii IV V vi vii* I
C D E F G A B C

The roman numerals above indicate a major or minor triad. The upper case are major and the lower case are minor. The vii* is a diminished triad. A * is used in place of a circle, which is most common. This pattern of major/minor/diminished applies to all major scales. Only chords I, IV, and V are major, ii, iii, and vi are minor and the vii* is diminished. A non-diatonic chord is any chord that does not belong to the key. For example, if a song is in G major.

I ii iii IV V vi vii* I
G A B C D E F# G

And a C minor chord is used, it would be a non-diatonic chord. Non-diatonic chords can be used to change the key of a song, but can also add interest to the harmony.
2b. Chord Progressions With Pedal Tones
A pedal tone is a note that is repeated or sustained while the harmony changes over it. Pedal tones are commonly used in progressive rock to add color to a chord progression. In the below example in 5/4, E is used as the pedal tone:
midi sample (Pedal Tones)

There are all kinds of ways to use pedal tones in chord progressions.
2c. Chord Progressions With A Moving Bass
A moving bass line is a great way to make a chord progression sound more melodic.
midi sample (Moving Bass)

In the example below, the bass notes emphasize the power chords in the first measure. In the second measure they lead the F# power chord to the Bb7 power chord. In the third measure the root and fifth are played to bring it all to a close. Using notes other then the root in the bass is common in bass lines and is called an inversion.

2d. Harmonic Modulation
Harmonic Modulation occurs when they key of the song is changed. Harmonic modulation is used a great deal in progressive music and gives the song variety and complexity.
Note: To understand this next section, a good solid background in diatonic harmony would be very helpful. Remember the formulas:

Notice that the only note difference between G major and A major are C# and G#? The C# and G#, in A major, alter the chords in G major. For example in G major, the A (ii) chord is minor because it is spelled A-C-E. In A major, the A chord is spelled A-C#-E, making it major. Another difference is in G major, the F# (vii*) chord would be diminished because it is spelled F#-A-C. In A major, it would be minor because it is spelled F#-A-C#. G major has an E(vi) minor chord, spelled E-G-B. A major has a E(V) major chord, spelled E-G#-B. G major has C(IV) and G(I) natural chords in it, but A major does not. A major has a C#(iii) minor chord in it and a G#(vii*) diminished.
Those are all the differences within the two keys, now look at the similarities. G major and A major both share a Bm and a Dm. NOTE: Many of the examples below are very simple and un-progressive. As an optional excercise, try to put these chord progressions into odd time signatures, while using a more creative rhythm pattern.
2e. Harmonic Modulation With Borrowed Notes
Using borrowed notes isn't a key change. It is simply a way to add non-diatonic chords to a diatonic melody. In the example below, the key will not change from G major. Instead, it will use some chords from A major. The below example is extremely simple in rhythm and time signature. As a nice progressive exercise, try to put this chord progression into an odd time signature, while using a more creative rhythm pattern.
midi file (Borrowed Chords)

2g. Harmonic Modulation With Pivot Chords
In a lot of music, a I chord comes after a V chord. Our ears are used to hearing this. So using the V chord as a pivot chord is a great way to get into a new key. The example below shows a harmonic modulation between G major and F major by using a pivot chord. In G major, the V chord is a DM. In F major, the V chord is a CM, but is only a IV chord in G major. So to make a C(IV) chord into a C(V) chord, we make it a dominant C chord. This gives it a feeling of being a V chord. The progression goes GM, DM, C7, FM. Of course, this example is very short and simple. Normally it would be longer to establish the keys.
midi sample (Pivot Chords)

2h. Harmonic Modulation With Relative Keys
In every major key there is a relative minor key that has the exact same notes in it, just in a different order. The best example is C major and A minor. They both have all natural notes in them, but sound very different.

Major: C D E F G A B C
Minor: A B C D E F G A

This is also how modes work. If the scale is started on a different note, it will be in a different mode. The major mode (C major) is also called the Ionian mode and the minor mode (A minor) is also called the Aeolian mode. Going to a relative minor/major key will not change the key of the song, so technically it's not harmonic modulation. It is, however, a very easy way to give a song a different feel. In the below example, G major will turn to E minor. Remember: The relative minor of any major key is the 6th degree of the scale and the relative major of any minor key is the 3rd degree of the scale.
midi sample (Relative Keys)

2i. Harmonic Modulation With Parallel Keys
Every key has a parallel key such as G major is parallel to G minor, C minor is parallel to C major, F# major is parallel to F# minor, etc. Changing a major key to a minor key or vise versa is called mutation. In the example below, E minor will mutate to E major.
midi sample (Mutation)

Final Thoughts

Hopefully you have learned a lot about harmony and time signatures in this lesson. If you found this lesson confusing, maybe you should consider taking music theory lessons. There is no good substitute for a teacher, so if your stuck, ask someone. Now, using the information you have learned, go write some music. The more you apply these concepts, the more you will understand them. Good luck, and keep creating original progressive music.

My next lesson will be about progressive melody and soloing concepts. Hopefully that will be done in the near future. And please don't PM me if you have questions.
My e-mail is blu9999@gmail.com if you have questions.

that was very well thought out and assembled. i don't see what was so complicated about it. i wonder about the whole math rock concept. if a song doesn't lend itself to an odd meter, why put it in there just to prove you can. just sayin. great lesson though.

ehh while there is good stuff here, you overcomplicate things, make things very dense with abstract examples and few practical comparisons (you mention math bands, so why not use some of their riffs to illustrate your points?) and you yourself, despite knowing all this theory, still write pretty boring songs

ehh while there is good stuff here, you overcomplicate things, make things very dense with abstract examples and few practical comparisons (you mention math bands, so why not use some of their riffs to illustrate your points?) and you yourself, despite knowing all this theory, still write pretty boring songs

It's your job to create the original and intersting music, not mine. I give you "guidelines" so you can create your own.

DOES ANY ONE KNOW TUNING??? I know that standard on a bass tuner iz no flats (E,A,D,G) and five flats are low B (B,E,A,D) and I know that two flats are (D,-,-,-,) somthin but what are 1,3,and4 flats called PLEEZ SOMEONE TELL ME!!!If u can THNX!!!